Simplify the following expression: $ r = \dfrac{-1}{3} - \dfrac{-5x - 2}{x - 8} $
Solution: In order to subtract expressions, they must have a common denominator. Multiply the first expression by $\dfrac{x - 8}{x - 8}$ $ \dfrac{-1}{3} \times \dfrac{x - 8}{x - 8} = \dfrac{-x + 8}{3x - 24} $ Multiply the second expression by $\dfrac{3}{3}$ $ \dfrac{-5x - 2}{x - 8} \times \dfrac{3}{3} = \dfrac{-15x - 6}{3x - 24} $ Therefore $ r = \dfrac{-x + 8}{3x - 24} - \dfrac{-15x - 6}{3x - 24} $ Now the expressions have the same denominator we can simply subtract the numerators: $r = \dfrac{-x + 8 - (-15x - 6) }{3x - 24} $ Distribute the negative sign: $r = \dfrac{-x + 8 + 15x + 6}{3x - 24}$ $r = \dfrac{14x + 14}{3x - 24}$